An adaptive scheme to generate reduced-order models for parametric nonlinear dynamical systems is proposed. It aims to automatize the POD-Greedy algorithm combined with empirical interpolation. At each iteration, it is able to adaptively determine the number of the reduced basis vectors and the number of the interpolation basis vectors for basis construction. The proposed technique is able to derive a suitable match between the reduced basis and the interpolation basis vectors, making the generation of a stable, compact and reliable reduced-order model possible. This is achieved by adaptively adding new basis vectors or removing unnecessary ones, at each iteration of the greedy algorithm. An efficient output error indicator plays a key role in the adaptive scheme. We also propose an improved output error indicator based on previous work. Upon convergence of the POD-Greedy algorithm, the new error indicator is shown to be sharper than the existing ones, implicating that a more reliable reduced-order model can be constructed. The proposed method is tested on several nonlinear dynamical systems, namely, the viscous Burgers' equation and two other models from chemical engineering.

中文翻译：

---

参数非线性动力系统的自适应基构造和改进的误差估计

提出了一种为参数非线性动力系统生成降阶模型的自适应方案。它旨在自动化结合经验插值的 POD-Greedy 算法。在每次迭代时，能够自适应地确定缩减基向量的数量和用于基构建的插值基向量的数量。所提出的技术能够在缩减基和插值基向量之间得出合适的匹配，从而使得生成稳定、紧凑和可靠的降阶模型成为可能。这是通过在贪婪算法的每次迭代中自适应地添加新的基向量或删除不必要的基向量来实现的。有效的输出误差指示器在自适应方案中起着关键作用。我们还基于以前的工作提出了一种改进的输出误差指标。在 POD-Greedy 算法收敛时，新的误差指标显示出比现有指标更清晰，这意味着可以构建更可靠的降阶模型。所提出的方法在几个非线性动力系统上进行了测试，即粘性伯格斯方程和其他两个化学工程模型。